Lehmer mean

A generalized mean that includes arithmetic, geometric, and harmonic means as special cases

The Lehmer mean is a type of generalized mean that is used in various fields such as mathematics, statistics, and signal processing. It is named after the American mathematician Derrick Henry Lehmer. The Lehmer mean provides a way to interpolate between different types of means, such as the arithmetic mean, geometric mean, and harmonic mean, by adjusting a parameter.

Definition

The Lehmer mean of a set of non-negative real numbers \( x_1, x_2, \ldots, x_n \) with a parameter \( p \) is defined as:

\[ L_p(x_1, x_2, \ldots, x_n) = \frac{\sum_{i=1}^n x_i^{p+1}}{\sum_{i=1}^n x_i^p} \]

where \( p \) is a real number. The parameter \( p \) determines the type of mean:

• For \( p = 0 \), the Lehmer mean is the Harmonic mean.
• For \( p = 1 \), the Lehmer mean is the Arithmetic mean.
• For \( p = -1 \), the Lehmer mean is the Geometric mean.

Properties

• Monotonicity: The Lehmer mean is monotonic in \( p \). This means that if \( p_1 < p_2 \), then \( L_{p_1}(x_1, x_2, \ldots, x_n) \leq L_{p_2}(x_1, x_2, \ldots, x_n) \).
• Homogeneity: The Lehmer mean is homogeneous of degree 1, meaning that for any positive scalar \( c \), \( L_p(cx_1, cx_2, \ldots, cx_n) = cL_p(x_1, x_2, \ldots, x_n) \).
• Symmetry: The Lehmer mean is symmetric in its arguments, meaning that the order of the elements does not affect the result.

Applications

The Lehmer mean is used in various applications, including:

• Signal processing: In signal processing, the Lehmer mean can be used to smooth signals or to compute averages that are less sensitive to outliers than the arithmetic mean.
• Statistics: In statistics, the Lehmer mean can be used to compute central tendency measures that are more robust to skewed data distributions.
• Economics: In economics, the Lehmer mean can be used to compute indices that account for different levels of income or wealth distribution.

Also see

• Generalized mean
• Power mean
• Weighted mean
• Minkowski inequality

   This article is a mathematics-related stub. You can help WikiMD by expanding it!